As we have seen in the previous subsection, in the univariate case,
\(\mathcal {F}_{\infty -\text{mars}}^1\)  consists of all the functions \(f\)  satisfying
(REF ) with some function \(g\)  having
finite total variation. An analogous characterization holds for all \(d\ge 1\) , and it will be presented in the next subsection. For general \(d\) ,
the role of total variation in \(d = 1\)  will be played by the Hardy-Krause
variation, which is an extension of total variation of
univariate functions to higher dimensions.
Thus, here we briefly introduce Hardy-Krause variation and recall some properties of the variation that we shall use in the following subsection.
Standard references for Hardy-Krause variation are [1]}
and [2]}.
